Research On Resistor Network Based On Special Matrix Model And Its Application

Resistor network modeling is a basic method to solve complex problems in engineering,physics,science and technology,and in real life,resistor network is also widely used.However,the network boundary conditions have strong complexity,high manual calculation cost and many restrictions,and it is difficult to calculate the potential formula of the complex resistor network with arbitrary boundaries.To solve the problem of limited calculation scale,this paper proposes a fast algorithm using numerical simulation method to improve the data calculation scale.The new potential formula and the proposed fast algorithm provide a new tool for science and engineering fields.Based on this,three kinds of special resistance networks are studied in this paper,and the main work is as follows:First,this paper introduces the research background and significance of this paper,expounds the research status of resistor network at home and abroad,introduces the relevant mathematical content and research methods according to the research background of this paper,and finally summarizes the research content and organizational structure of this paper.Secondly,the second kind of Chebyshev polynomial is introduced to represent the analytical solution of the resistor network,and the potential formula of the hammock resistor network is improved.Then,a fast algorithm is given by using the second kind of discrete cosine transform(DCT-Ⅱ)and matrix vector multiplication.In application,the equivalent resistance formula given in special cases is displayed through 3D view.Finally,comparing the computational efficiency of the original formula,the improved formula and the fast algorithm,the improved potential formula and the proposed fast algorithm achieve large-scale operations.Thirdly,a cylindrical resistor network model with complex boundary and its potential formula are proposed.The resistor network model is composed of Kirchhoff’s law,based on the Recursion Transform-I(RT-I)method and the quasi-triagonal Toeplitz matrix.The main method is to construct the orthogonal matrix transformation to obtain the eigenvalues of the quasi-triagonal Toeplitz matrix,and use the fifth kind of discrete sine transformation(DST-Ⅴ)to obtain the node current.The potential formula expressed by Chebyshev polynomials is proposed.In addition,potential formulas under some special parameters are also considered and displayed by using 3D views.Finally,a fast algorithm for calculating electric potential is given by using the constructed resistor network matrix equation model,DST-Ⅴ and fast matrix vector multiplication.Fourth,a horn toroidal resistance network model with special boundary and its electric potential formula are proposed.According to Kirchhoff’s law and the Recursion Transform-V(RT-V)method,a new resistor network model is established by using the voltage V and the perturbed tridiagonal Toeplitz matrix,and the exact electric potential formula of the horn torus resistor network is obtained.Firstly,the eigenvalues and eigenvectors of the perturbed tridiagonal Toeplitz matrix are obtained by orthogonal matrix transformation;Secondly,the solution of node voltage is given by DST-Ⅴ,and the exact potential formula is expressed by Chebyshev polynomial.In addition,the equivalent potential formula in special cases is given and displayed in 3D view.Finally,using the new mathematical model,DST-Ⅴ and matrix vector fast multiplication,a fast algorithm for calculating the resistor network potential is given.Fifthly,it summarizes the work and innovation of the full text,and points out the shortcomings in the current work and the future research direction.